The equivalent capacitance of the combination shown is

In the given circuit, the charge on \(4\, \mu F\) capacitor will be.....\(\mu C\)

A thin convex lens made from crown glass \(\left( {\mu = \frac{3}{2}} \right)\) has focal length \(f\). When it is measured in two different liquids having refractive indices \(\frac{4}{3}\) and \(\frac{5}{3}\) it has the focal lengths \(f_1\) and \(f_2\) respectively. The correct relation between the focal lengths is:

A passenger train of length \(60\, m\) travels at a speed of \(80\, km/hr\). Another freight train of length \(120\, m\) travels at a speed of \(30\, km/hr\). The ratio of times taken by the passenger train to completely cross the freight train when: \((i)\) they are moving in the same direction, and \((ii)\) in the opposite directions is

The binding energy per nucleon of \(^{209}_{83}Bi\) is _______ MeV.
\([\text{Take } m(^{209}_{83}Bi) = 208.980388 \, u, \, m_p = 1.007825 \, u, \, m_n = 1.008665 \, u, \, 1 \, u = 931 \, \text{MeV}/c^2]\)

Four particles each of mass \(1 \mathrm{~kg}\) are placed at four corners of a square of side \(2 \mathrm{~m}\). Moment of inertia of system about an axis perpendicular to its plane and passing through one of its vertex is_________ \(\mathrm{kgm}^2\).

A thin circular disc is in the \(xy\) plane as shown in the figure. The ratio of its moment of inertia about \(z\)  and \(z'\)  axes will be

Three conductors of same length having thermal conductivity \(\mathrm{k}_1, \mathrm{k}_2\) and \(\mathrm{k}_3\) are connected as shown in figure.

Area of cross sections of \(1^{\text {st }}\) and \(2^{\text {nd }}\) conductor are same and for \(3^{\text {rd }}\) conductor it is double of the \(1^{\text {st }}\) conductor. The temperatures are given in the figure. In steady state condition, the value of \(\theta\) is _______ \({ }^{\circ} \mathrm{C}\).
(Given : \(\mathrm{k}_1=60 \mathrm{Js}^{-1} \mathrm{~m}^{-1} \mathrm{~K}^{-1}, \mathrm{k}_2=120 \mathrm{Js}^{-1}\mathrm{~m}^{-1}\) \( \mathrm{~K}^{-1}, \mathrm{k}_3=135 \mathrm{Js}^{-1} \mathrm{~m}^{-1} \mathrm{~K}^{-1}\) )

If for \(a >0,\) the feet of perpendiculars from the points \(A ( a ,-2 a , 3)\) and \(B (0,4,5)\) on the plane \(l x+m y+n z=0\) are points \(C(0,-a,-1)\) and \(D\) respectively, then the length of line segment \(CD\) is equal to

The threshold wavelength for photoelectric emission from a material is \(5500\,\mathring A\). Photoelectrons will be emitted, when this material is illuminated with monochromatic radiation from a \(A.\) \(75\,W\) infra-red lamp \(B.\) \(10\,W\) infra-red lamp \(C.\) \(75\,W\) ultra-violet lamp \(D.\) \(10\,W\) ultra-violet lamp Choose the correct answer from the options given below :

Let \(a\), \(b\) be two non-zero real numbers. If \(p\) and \(r\) are the roots of the equation \(x ^{2}-8 ax +2 a =0\) and \(q\) and \(s\) are the roots of the equation \(x^{2}+12 b x+6 b\) \(=0\), such that \(\frac{1}{ p }, \frac{1}{ q }, \frac{1}{ r }, \frac{1}{ s }\) are in A.P., then \(a ^{-1}- b ^{-1}\) is equal to \(......\)

A body of mass \(m= 10^{-2}\) \(kg\) is moving in a medium and experiences a frictional force \(F= -kv^2\). Its initial speed is \(v_0= 10\) \(ms^{-1}\). If, after \(10\; s\), its energy is \(\frac{1}{8}\) \(mv_0^2\) the value of \(k\) will be

The radius in kilometer to which the present radius of earth \(( R =6400\, km )\) to be compressed so that the escape velocity is increased \(10\) time is ............\(km\)